CAT 2025 Lesson : Functions - Domain, Codomain & Range

2.1 Domain, Co-domain and Range

Brief definitions of the three are as follows.
$1$) Domain is the set of all possible inputs or $x$-values.
$2$) Co-domain is the set of possible outputs or values of $f(x)$ that is defined by us.
$3$) Range is the set of outputs that are actually generated for the given inputs or domain. These are the actual values that $f(x)$ can possible take.

The following diagram is a pictographic representation of $f(x) = |x|$, where $x$ is an integer between $-2$ and $2$ (both inclusive). And, the co-domain has been defined as {$-2, -1, 0, 1, 2$}.

As $f(x)$ is never negative, the range does not include $-2$ and $-1$ and contains only the non-negative integers.



Two other classifications of functions are as below.
$1$ Onto Functions: Every element in the co-domain is related to at least one value in the domain. In this case, the co-domain is the range.
$2$ Into Functions: There is at least one element in the co-domain that is not related to a value in the domain. In this case, while the range is a subset of the co-domain, it has fewer values than the co-domain.


Reminder: As discussed earlier, one-to-many and many-to-many relationships are not functions. As shown earlier, $y^2 = x$ or $y = \pm \sqrt{x}$, where $x \ge 0$ is not a function as each $x$-value (other than $0$) results in $2 \space y$-values.

3. Common Types of Functions

Some common types of functions are given below, with examples.

Linear Functions are of the form $y = mx + c$. This results in a straight line. Adjacent is the graphical representation of $f(x) = 0.5x - 2$ Type: One-to-One function Domain: Set of all real numbers Range: Set of all real numbers
Quadratic Functions have a U-shaped or an inverted-U shaped curve, as discussed in the Quadratic Equations lesson Adjacent is the graphical representation of $f(x) = x^2 - 4$ Type: Many-to-One function Domain: Set of all real numbers Range: Set of all real numbers greater than or equal to $-4$
In Cubic Functions, the highest power of the polynomial is $3$. Adjacent is the graphical representation of $f(x) = x^3$ Type: One-to-One function Domain: Set of all real numbers Range: Set of all real number
In Bi-quadratic Functions, the highest power of the polynomial is $4$. Adjacent is the graphical representation of $f(x) = x^4$ Type: Many-to-One function Domain: Set of Real numbers Range: Set of real numbers greater than or equal to $-1$.
Rational Functions are those where the numerator and denominator contain a polynomial. The function is not defined when the denominator equals zero. Adjacent is the graphical representation of $f(x) = \dfrac{2}{x - 1}$ Type: One-to-One function Domain: Set of real numbers other than $+1$. Range: Set of real numbers other than $0$.
The Modulus Function of a number provides the non-negative value of the number. To define it as a function, $f(x) = |x| = x \space \text{if} \space x \space \ge 0$          $= -x \space \text{if} \space x < 0$ The above function is represented as $f(x) = |x|$ Type: Many-to-One function Domain: Set of all real numbers Range: Set of all non-negative real numbers